richyw

4 years ago

Compute the polar moment of inertia \(I_0\) for a disc \(D\) centered at the origin with radius \(a\) and constant density \(\rho (x,y)=\rho\). Write (\I_0\) in terms of the radius \(a\) and the total mass \(m\) of \(D\)

Compute the polar moment of inertia \(I_0\) for a disc \(D\) centered at the origin with radius \(a\) and constant density \(\rho (x,y)=\rho\). Write \(I_0\) in terms of the radius \(a\) and the total mass \(m\) of \(D\)

I'm not sure at all how to do this. I have been working on it for like 6 hours and feel I am getting nowhere. No questions like this in my textbook. I could do all those ones fine (in about 1/4 that time haha)

oh wait I'm opening this again. I just noticed when I did the first part. I got a different answer. This is what I did.\[I_0=\iint_R(x^2+y^2)\rho(x,y)dA\]\[=\rho \int^{2\pi}_0\int^a_0(r^2\cos^2\theta+r^2\sin^2\theta)\,r\,dr\,d\theta\]\[=\rho \int^{2\pi}_0\int^a_0r^2 (cos^2\theta+\sin^2\theta)\,r\,dr\,d\theta\]\[=\rho \int^{2\pi}_0\int^a_0r^3\,dr\,d\theta\]\[=\frac{\pi \rho a^4 }{2}\]